Finale of THIS.
Unicode says that $\Cup$ and $\Cap$ are double union and intersection, respectively. I was wondering if there was an actual operation that went with these symbols. If not, would these definitions make sense for these operators? As follows:
$$A\Cup B:=\left\{(x,x):x\in(A\cup B)\right\}$$
and
$$A\Cap B:=\left\{(x,x):x\in(A\cap B)\right\}$$
Question
Do these operators exist within Set Theory? Iff not, do they exist anywhere in the realm of mathematics? Is my idea for these two operators logical and useful?
Best Answer
I think a more useful definition would be $A\Cup B:=\{a\cup b\mid a\in A, b\in B\}$, respectively $A\Cap B:=\{a\cap b\mid a\in A, b\in B\}$. I've never seen the symbol before, but I could think of situations where that might be useful. I'm sure there are situations where the diagonal of $(A\cup B)^2$ (which you're using) is used, but I have no idea, when you'd want to introduce a special symbol for that, and why you'd want to pick one, that looks more like a union than a diagonal, which is usually denoted by $\Delta$...